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Time-reversal-broken Weyl semimetal in the Hofstadter regime

by Faruk Abdulla, Ankur Das, Sumathi Rao, Ganpathy Murthy

This Submission thread is now published as SciPost Phys. Core 5, 014 (2022)

Submission summary

As Contributors: Faruk Abdulla · Ankur Das · Ganpathy Murthy · Sumathi Rao
Arxiv Link: (pdf)
Date accepted: 2022-02-14
Date submitted: 2022-01-12 17:54
Submitted by: Abdulla, Faruk
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


We study the phase diagram for a lattice model of a time-reversal-broken three-dimensional Weyl semimetal (WSM) in an orbital magnetic field $B$ with a flux of $p/q$ per unit cell ($0\le p \le q-1$), with minimal crystalline symmetry. We find several interesting phases: (i) WSM phases with $2q$, $4q$, $6q$, and $8q$ Weyl nodes and corresponding surface Fermi arcs, (ii) a layered Chern insulating (LCI) phase, gapped in the bulk, but with gapless surface states, (iii) a phase in which some bulk bands are gapless with Weyl nodes, coexisting with others that are gapped but topologically nontrivial, adiabatically connected to an LCI phase, (iv) a new gapped trivially insulating phase (I$'$) with (non-topological) counter-propagating surface states, which could be gapped out in the absence of crystal symmetries. Importantly, we are able to obtain the phase boundaries analytically for all $p,q$. Analyzing the gaps for $p=1$ and very large $q$ enables us to smoothly take the zero-field limit, even though the phase diagrams look ostensibly very different for $q=1, B=0$, and $q\to\infty, B\to 0$.

Published as SciPost Phys. Core 5, 014 (2022)

Author comments upon resubmission

We thank all the referees, especially referee 3, for their detailed and contructive comments which have helped us to vastly improve the manuscript. We have extensively revised the introduction and the summary, by making more explicit the novelty and context of our work, emphasized the new phases found, and discussed their implications in detail.

List of changes

1. We have completely revised the abstract, making sure that new results are given due prominence.

2. We have thoroughly reworked the introduction in accordance with the suggestion of Referee 3. Now we provide a detailed explanation of the precise way in which our work extends previous results (minimal crystalline symmetry, field perpendicular to the line separating the Weyl nodes), and the importance of the semiclassical limit. We also list the phases we see, including the W2’ and I’ phases which are only seen at nonzero field.

3. Despite the minimal crystalline symmetry, there are a number of symmetries enjoyed by the zero-field Hamiltonian. For completeness, we have devoted an appendix to listing all these symmetries.

4. In accordance with Referee 3’s suggestion that we show the reader how the surface states decay into the bulk for the Fermi arc states and the surface states corresponding to the LCI, we have added four panels in Fig. 3, and an entirely new Fig. 4.

5. In subsection 3.3.3, we have added a paragraph to describe a topological response of W2′ phase, which is additive between the WSM surface states and the LCI surface states.

6. We have revised 3.3.4 describing the I′ phase. We offer a natural way to think about it as arising from the W2’ phase, to which it is always adjacent in the phase diagram.

7. The panels in Fig. 6 have been reorganised for clarity of presentation.

8. Panel 9c has been updated, and now shows the gap rather than its logarithm.

9. Two new references (33 and 34) concerning the annihilation of Weyl nodes in a magnetic field have been added in section 5.

10. In accordance with Referee 3’s suggestion, in the summary, the paragraphs about disorder and interactions have been completely revised, and references added.

11. A short paragraph on potential ways of experimentally realizing the model we use has been added in the summary.

Reports on this Submission

Anonymous Report 2 on 2022-2-3 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2108.03196v3, delivered 2022-02-03, doi: 10.21468/SciPost.Report.4311


see report


see report


I thank the authors and already said the results are what they are and that I can endorse publication as reference for a calculation involving this model. However for the record I do like to state the following;

1. We agree that the simpler the model the better to illustrate the physics. So indeed the toy model for 2 nodes in the bulk has since the Nielsen-Ninomiya work,, been very important, but such a toy model should stand at the basis of a universal new effect, e.g. Weyl excitations giving rise to a chiral anomalous effect. Just calculating the phase diagram of a toy model is not a result that induces much novelty. Furthermore the effects the authors discuss do start from a model that needs to match the experimental situation [this model of nodes at the Fermi level, these symmetries] and the surface cut, see also point 4.

2. Here I like to point out that quite generally the relation to stacked Chern insulators and thus TRS breaking slice in momentum space [c.f. Also role of magnetic field as we know from e.g. Haldane model that TRS breaking terms can induce Chern layers] is rather well known. Indeed it is a manner to visualize Weyl states see Fig 1 in, so when under magnetic field when one gets transitions the results are to be expected from these general point of view. The authors point out arXiv:2105.08443 but there is quite extensive literature on Weyl semimetals in magnetic fields, and with regard to their comment on the crystalline symmetry I refer to point 4 and 5.

3. We agree about this but this is also tied to simplicity of the model.

4. The definition of minimal crystalline symmetries is space group 1, i.e. P1 having no inversion [P1'] or two fold rotations. Due to the simplicity of the model [only nn hopping, 2 band] this is not realized and should be taken into account with their own comments and comparison to other papers in point 2, see my point 5.

5. Already from a simple point of view it is puzzling why the authors claim novelty or minimal crystalline symmetries. Indeed, not wanting to start a discussion of how it is not surprising that a 2-band Weyl model will transition of course via Weyl points to possible layered Chern insulators etc or other Weyl phases or gapped phases, we can consider the authors' own reasoning in point 2. They state that arXiv:2105.08443 is different as it needs spatial symmetries. But if their system had no crystalline symmetries or only translations how is it possible that they find a gapped phase with protected edge states? Indeed, this is because of point 4. So discarding the fact that general reasoning shows that the authors' results are not surprising, even in their own context their arguments are confusing.

  • validity: ok
  • significance: ok
  • originality: low
  • clarity: high
  • formatting: perfect
  • grammar: perfect

Anonymous Report 1 on 2022-1-27 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2108.03196v3, delivered 2022-01-27, doi: 10.21468/SciPost.Report.4255


- improved clarity
- better storyline than before


Inherent to the paper is a very narrow focus and lack of new insights


I thank the authors for their reply.

The manuscript has improved and in principle it can be endorsed as a result.

Overall the response to my questions was satisfactorily. I however repeat that there are not many new insights. With my previous questions on novelty and motivation I intended to spark a discussion on this topic. The authors are correct that they analyze a slightly different problem compared to the references in the introduction. This was however not the idea of my comment. The point is that the authors consider a simple toy model for Weyl nodes [[in fact so simple that not all symmetries can be broken, due to presence of 2-band nn model]] and the introduction of the field not suprisingly invokes Weyl nodes and transitions. Such consequences of this effective Weyl model have been well considered in a very vast body of literature. So given that new general insights/principles are obtained, such a consideration is what it is, but at least narrow in terms of impact and new insight or applicability to experiment. Indeed also the crystalline phase is inherent to the toy model and for experiment will need the right symmetries and will suffer from the complication that this effective model for the Weyl nodes at the Fermi level will not capture any realistic system, especially in terms of edge states.

  • validity: ok
  • significance: ok
  • originality: low
  • clarity: high
  • formatting: perfect
  • grammar: perfect

Author:  Faruk Abdulla  on 2022-02-02  [id 2147]

(in reply to Report 1 on 2022-01-27)
reply to objection

We thank the referee for finding our manuscript to be improved with better presentation. However, in the referee's opinion, there is not enough novelty in our paper. Furthermore, the referee feels that the model is too simple to be applied to experiments. While it is difficult to argue with an opinion, we respectfully disagree, and offer the following rebuttal.

  1. It is true that our model is simple, but we consider this a virtue. It is often easier to see qualitatively new results in a simple model than in one with all the complexities of a particular experiment. The simplicity of the model allows us to fully characterize the phase diagram for arbitrary commensurate fields, which is itself a new result.

  2. We also find two new phases, one which displays the coexistence of Weyl nodes and attendant surface Fermi arcs, and the surface conduction of a Layered Chern insulator. The only other example of this phenomenon currently known to us, arXiv:2105.08443, needs special symmetries. This is new and has direct experimental consequences.

  3. The fact that we can analytically find the phase diagram allows us to approach the semiclassical limit systematically, which is a new result.

  4. We are not quite sure what the referee means when they claim that the model is too simple to break all the symmetries. We have in fact considered the case with minimal lattice symmetry. We therefore expect the results to be more generic than those obtained in a model with a higher symmetry.

We appreciate the comments by the referee and would like to point out that referee three in the previous round found our results appealing.

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